论文标题
加权均匀空间的几何形状
Geometry of Weighted Homogeneous Spaces
论文作者
论文摘要
在本文中，我们定义了加权均质空间（WHS），由$ \ frac {g} {p} [ψ_h] $表示，其中$ψ_H$是$ g $的简单根部定义的$ g $的权重函数，由元素$ h $在最高的Weyl Chamber中。权重函数$ψ_h$描述了最大圆环$ t $在不同的bruhat单元上的作用，并且通过由weyl oft $ w $的动作定义的坐标的变化来表现良好。本文中的主要努力是证明加权均匀空间的基本代数和几何特性。可以将定义与Reid-Corti \ cite {Cr}给出的现有版本进行比较。此外，我们通过有限的Abelian组的某些操作来表达$ \ frac {g} {p} [ψ_h] $作为$ g/p $的整个紧凑商。此外，当两个具有不同重量系统的WH是同构时，它是一个标准。该标准提供了一种简单的方法，可以理解两个WH的常规图，该图由具有特定多项式条目的矩阵定义。我们还通过在$ g $上使用某些潜在功能来解释WHS上不变的Kähler差异。我们的贡献是对\ cite {al，al，akq}的结果的概括。为此，我们解释了权重如何影响\ cite {al，akq}中的Chern类别捆绑包的不同计算。最后，我们通过与加权箭弹相关的群集代数为WHS的坐标环提供了结果。具体而言，我们表明WHS的坐标环是有限类型的加权群集代数。在这种情况下，相应的Dynkin颤动配备了在突变也影响权重的顶点上定义的重量函数。我们将WHS的嵌入在加权射击空间的产物中，表明坐标环是加权分级代数。
In this paper, we define the weighted homogeneous space (WHS), denoted by $\frac{G}{P}[ψ_H]$ where $ψ_H$ is weight function defined on the set of simple roots of $G$, by an element $H$ in the highest Weyl chamber. The weight function $ψ_H$ describes the action of the maximal torus $T$ on different Bruhat cells and is well behaved via the change of coordinates defined by the action of the Weyl group $W$. The major effort in this text is to prove basic algebraic and geometric properties of a weighted homogeneous space. The definition can be compared with an existing version given by Reid-Corti \cite{CR}. Additionally, we express $\frac{G}{P}[ψ_H]$ as a whole compact quotient of $G/P$ by a certain action of a finite abelian group. Besides, it is presented a criterion when two WHS with possibly different weight systems are isomorphic. The criteria give a simple method to understand the regular maps between two WHS's, defined by matrices with specific polynomial entries. We also explain invariant Kähler differentials on WHS by using certain potential functions on $G$. Our contribution is a generalization of the results presented in \cite{Al, AL, AKQ}. For that, we explain how the weights affect different computations of chern classes of line bundles given in \cite{AL, AKQ}. Finally, we provide a result on the coordinate ring of a WHS by cluster algebras associated to weighted quivers. Specifically, we show that the coordinate ring of a WHS is a weighted cluster algebra of finite type. In this case, the corresponding Dynkin quiver is equipped with a weight function defined on the vertices where the mutations also affect the weights. We present an embedding of a WHS in a product of weighted projective spaces, showing that the coordinate ring is a weighted graded algebra.
